Friday, August 12, 2016

If Buridan's contribution to the bestiarum philosophorum was the ass, and David Armstrong's the ostrich, Arianna Betti's is the hedgehog bristling with spines. The hedgehog is an appropriate totemic animal inasmuch as her book too bristles with sharp distinctions and prickly arguments designed to inflict pain upon the friends of facts. In this penetrating and beautifully organized volume Professor Betti deploys her distinctions and arguments against two sorts of facts, compositional and propositional, as she calls them. The states of affairs of David Malet Armstrong's middle period (Armstrong 2007) are examples of the first kind of fact. These items are the main target of Betti's animadversions in the first part of her two-part book. She does not go so far as to claim that Armstrongian facts do not exist; her claim is the rather more modest one that we have no reason to posit them, since the work they do, if it needs doing at all, can be done just as well by a certain sort of mereological sum. (101) Betti ignores, however, Armstrong's very different later conception of states of affairs or facts. (Armstrong 2009; Armstrong 2010, 26-34; Vallicella 2016) This later conception also counts as compositional in her sense and ought to have been discussed for the sake of completeness, especially since it in some ways approximates to Betti's mereological position.

One might wonder how a fact could fail to be compositional. Facts are complex or composite items, after all, not simples. So they must all have some internal composition or other, whether they be truthmaking facts or facts of the Chisholmian-Plantingian sort. At a bare minimum, a's being F is composed of a and F-ness. Thus I find less than felicitous Betti's talk of propositional facts in contrast to compositional facts as “noncompositional objects at the level of reference.” (24) She makes it clear, however, that she is using 'compositional' in a narrow sense that implies that compositional facts and their constituents are “part[s] of the furniture of the world.” (37) We shall soon see that being in the world involves being real as opposed to being ideal. An example of a compositional fact is the fact of Guido's being hungry. This fact has Guido himself, all 200 lbs of him, as a constituent. An example of a propositional fact is the putative referent of the that-clause in a sentence like 'Guido sees that Francesca is serving spaghetti puttanesca.' This putative referent is the fact that Francesca is serving spaghetti puttanesca. This propositional fact is like a (Fregean) proposition, though it is not a proposition, in that it does not have Francesca herself as a constituent, but rather an abstract surrogate that represents her. (170) (This fact-of vs. fact-that terminology is mine, not Betti's. I got it from Milton Fisk.)

Betti describes in marvellous detail seven features of compositional facts (18) and five of propositional facts (170). I will speak of C-facts and P-facts. Here are some salient differences. C-facts are in the world, and thus suited to play the truthmaking role whereas P-facts are not in the world and hence not fit for truthmaking. To be in the world is to be real where to be real is to exist “through time and in time as causes or effects in a causal chain.” (22) So C-facts are real while P-facts are ideal. The ideality of P-facts, however, is not that of propositions since P-facts are not propositions. Betti is greatly and rightly exercised by the curious in-between status of these “ghostly critters” (114) that are neither truthbearers nor truthmakers and yet are championed by such distinguished philosophers as Roderick Chisholm, Alvin Plantinga, and Kit Fine. These “ghostly critters” are not truthbearers because they are neither true nor false. But while they are not bivalent in terms of truthvalue, they are 'bipolar' (my term): while all exist, some of them obtain while some do not. They are not truthmakers since truthmakers are real and 'monopolar': if they don't exist they are nothing. Thus the fact of Guido's being hungry does not exist at all if Guido is not hungry. Propositional facts are neither fish nor fowl. The conclusion Betti arrives at strikes me as correct: “Propositional facts collapse into true propositions.” (179) Propositional facts are thus not a distinctive category of entity. We need them, she thinks, as little as we need compositional facts. Actually, her position is far more radical than this since she denies that that-clauses are referential parts of speech. So her position is best expressed conditionally by the following quotation: “If there were nominal reference to facts, facts would be true propositions . . . . (113) Her view, if I understand it, is eliminativist not identitarian: she is not saying that there are propositional facts and that what they are are true propositions; she is saying that that there are no propositional facts.

Leaving propositional facts to languish in their ghostly realm, the rest of this article will take issue with Betti's critique of compositional facts, the ones dear to my heart, the facts involved in the flux and shove of the real order. On a personal note, I want to thank Professor Betti for her very close attention to my articles on the topic.

The Case Against Compositional Facts

A compositional fact, as opposed to a propositional fact, is an entity fit to play the role of truthmaker. The truthmaker role may be introduced as follows. Consider the assertive utterance of some such contingent sentence as 'Tom is sad.' If true, this assertively uttered sentence cannot just be true: if true, it is true because or in virtue of something external to it. This use of 'because' is not causal which is why philosophers reach for the weasel phrase 'in virtue of,' which, despite its slipperiness, may well be indispensable for metaphysics. I say it is indispensable. (Or do hedgehogs eat weasels?) Roughly, there has to be something that 'makes' the sentence true. This external something cannot be another declarative sentence, even if true. More generally, a truth is a true truthbearer (a Fregean proposition, say, or perhaps an Aristotelian proposition, see pp. 31-32 for Betti's helpful explanation of the difference) and no true truthbearer is made true by another such item in the specific sense of 'makes true' in play in truthmaker theory. Nor can someone's say-so be what makes true a true truthbearer. The truthmaker has to be something 'in the world,' something extralinguistic and mind-independent in the realm of reference as opposed to the realm of sense. The friends of truthmakers are realists about truth: they are convinced that at least some truths are in need of an ontological ground of their being true.1

Truthmaker maximalists hold that all truths need such grounds, but one needn't be a maximalist to be a truthmaker theorist. As for 'makes true,' this is neither entailment nor causation. Not entailment, because entailment is a relation between propositions, assuming that truthbearers are propositions, whereas truthmaking is a relation between extra-propositional reality and propositions. So if x makes true y, then y is a truthbearer, but x is not. If someone says that the proposition expressed by 'Snow is white' makes true the proposition expressed by 'Something is white,' then that person, while talking sense, is not using 'makes true' in the specific way in which the phrase is used in truthmaker theory. Truthmaking is not causation for a similar reason: causation does not connect the extra-propositional to the propositional whereas truthmaking does. As Armstrong says, truthmaking is “cross-categorial.” (Armstrong 2004b, 5) It links the extra-propositional to the propositional.

It is important to note, however, that while truthmakers cannot be Fregean or Aristotelian propositions, and thus must be extra-propositional, they must also be proposition-like on Armstrong's approach. This is a point I think Betti misses. Speaking of compositional facts, she tells us that “facts are neither linguistic nor languagelike entities at the lowest level of reference. (28, emphasis in original) But this is certainly not Armstrong's view, the view that is supposed to be the target of Betti's critique of compositional facts. His view is that the world is a world of states of affairs, a “totality of facts not of things” (Wittgenstein) and “sentence-like rather than list-like.” (Armstrong 2010, 34) If the world is sentence-like, then, pace Betti, it is language-like. Armstrong was profoundly influenced by his teacher in Sydney, the Scots philosopher John Anderson, who held that “reality, while independent of the mind that knows it, has a 'propositional' structure.” (Armstrong 1997, 3) Armstrong goes on to say that “the propositional view of reality which he [Anderson] championed is the facts or states of affairs view of reality.” (Armstrong 1997, 3-4) That Armstrongian facts are proposition-like and thus language-like is fairly obvious when we consider the truthmakers of contingent predications of the form 'a is F.' The truthmaker cannot be a by itself, or F-ness by itself, or the mereological sum a + F-ness. It must be a-instantiating F-ness, which has a proposition-like structure. Armstrongian facts have a logos-like and thus logical articulation contrary to what Betti says in opposition to Kit Fine. (28) But now I am getting ahead of myself.

Suppose you accept the legitimacy of the truthmaker role and the need for some type of entity to play it. It doesn't follow straightaway that the entities needed to play the role must be what Betti calls compositional facts or what David Armstrong calls states of affairs. This is so even if we confine ourselves to the really clear examples of truthbearers in need of truthmakers, namely, synthetic, contingent predications such as 'Guido is hungry' or the propositions expressed by assertive utterances of such sentences. Nevertheless, a powerful argument can be mounted for compositional facts as truthmakers. The argument Armstrong and I consider powerful, however, Betti calls “unsound.” (106) Surprise!

Although she is skeptical of the need for truthmakers, she is willing to grant the need arguendo, insisting only that if we need truthmakers, a certain type of mereological complex can do the job thus rendering Armstrong's facts, as unmereological complexes, unnecessary. (102) This is why she thinks the truthmaker argument for Armstrongian facts is unsound. As she sees it, compositional facts are not givens, but theoretical posits, and unnecessary ones at that. They were invented to solve a problem, the unity problem, that arises only because of certain optional assumptions about relations and properties that one is not bound to make. (94-95) Compositional facts are an ad hoc, indeed a “maximally ad hoc,” solution to a pseudo-problem. (64)

Now let me say something in exposition of Armstrong's argument for facts or states of affairs as truthmakers on the assumption that the truthmaker role is legitimate and needs to be filled by some category of entity or other. I will then consider Betti's counter-proposal.

If it is true that Tom is sad, could the truthmaker of this truth be the item that Betti calls (8) the sentence-subject of 'Tom is sad,' namely, Tom? No, since Tom needn't be sad. So Tom by himself cannot be what makes true 'Tom is sad.' The same goes for the property of being sad. By itself the property cannot be the truthmaker of the sentence in question. (I am assuming, with Armstrong, that properties are immanent universals. Immanent, in that they cannot exist uninstantiated; universal, in that they are repeatable.) Now if Tom exists and sadness exists, then so does the mereological sum Tom + sadness. But this sum cannot be the truthmaker either. For the sum exists whether or not Tom is sad. How so?

Suppose that Tom is not sad, but Shlomo is. If properties are immanent universals, then sadness cannot exist uninstantiated; suppose it exists in virtue of being instantiated by Shlomo. So Tom exists, sadness exists, and their sum exists. But this does not suffice for Tom's being sad. There is a missing ontological ingredient: something to connect sadness to Tom. You might think that the missing ingredient would have to be the worldly correlate of the 'is' of predication. But if you take this correlate to be an exemplification/instantiation relation then you ignite Bradley's relation regress which is unfortunately vicious. Other moves invoking Strawsonian nonrelational ties, Bergmannian nexus, Fregean unsaturated concepts, and benign fact-internal infinite regresses (see Vallicella 2010), are equally unavailing. The unifier of a fact's constituents cannot be a further constituent or anything internal to the fact. This leaves two possibilities: (i) the unifier is external to the fact, which Betti rejects, and (ii) Armstrong's middle-period suggestion that facts are entities in addition to their consituents and it is they who hold fact-appropriate constituents together so that they can exercise the truthmaking function. Betti has mastered the dialectic and considers the least bad solution to be Armstrong's: facts hold their constituents together. Although she doesn't say so, she considers my solution in terms of an external unifier to be the worst. The extant putative solutions to the unity problem of course presuppose that it is a genuine problem. Betti thinks it isn't.

Betti's Dissolution of the Unity Problem

After rejecting the extant putative solutions to the unity problem, Betti proposes to dissolve it by collapsing the distinction between “relations that relate relata and relations that do not: all relations relate relata and carry out their own unifying work.” (95) She means this to apply to properties as well. All properties qualify their bearers and carry out their own qualifying work. Thus there needn't be anything to hold the constituents of a relational or as monadic fact together: nothing internal to the fact, nothing external, and not the fact itself. Betti's point is that there is no need for Armstrongian facts, facts as entities in addition to their constituents. (Cf. Armstrong 1997, 117) Her point is not that there are no facts. There may well be facts; it is just that if there are, they are a special sort of mereological sum. Perhaps we can say that she is an identitarian about compositional facts, not an eliminativist, whereas she is an eliminativist about propositional facts, not an identitarian. More on this in a moment.

What Betti has to do is block a possibility like the following. In the actual world, call it Charley, Tim loves Tina. In a merely possible world w in which Tim and Tina both exist, Tim does not love Tina, but Tim loves Toni. In Charley we have both the relational fact of Tim's loving Tina and the mereological sum Tim + loves + Tina. In w, we have the sum Tim + loves + Tina but not the corresponding fact. This implies that there is more to the fact than the sum of its constituents: the sum can exist without constituting a fact. The something more is that which makes of the constituents a real truthmaking unity. Call it the unifier. Betti thinks that the least bad of the extant proposals as to what the unifier is is Armstrong's: facts hold their constituents together; facts are unmereological complexes over and above their constituents. In short, what Betti needs to do is counter the seductive thought that in an actual relational situation such as that of Tom's loving Tina, the constituents can exist without forming a real truthmaking unity. What she needs to maintain is that, necessarily, if all the constituents exist, then the relatedness exists. If the mere existence of the constituents ensures their connectedness, then there is no need for Armstrongian facts. You would then have real unity on the cheap, real truthmaking unity from mereology alone, or rather from mereology operating upon the right sorts of constituents. The mereological principle of the extensionality of parthood would hold for all complexes. Nice work if you can get it!

Betti can achieve her end if she holds that relations are relata-specific where “A relation is relata-specific if and only if it is in its nature to relate specific relata.” (89) Suppose that the relation loves as it figures in the sum Tom + loves + Tina is necessarily such that, if it exists, then it relates Tom and Tina. Then there would be no distinction in reality between loves as a relating relation and loves as an inert relation that is merely a constituent but not also a unifier of the complex into which it enters.

Betti's contention, then, is that all relations, just in virtue of existing, are relating relations, active ontological ingredients if you will, and none are inert ingredients. A relation cannot exist without actually relating its relata. If so, there cannot be a difference between the mereological sum a + R+ b and the fact of a's standing in R to b. Given the constituents, the fact is given: it is not an ontological extra, something over and above the constituents. There is no possibility of the constituents existing without the fact existing. It follows that there is no need for facts as unmereological compositions, facts as “additions to being,” in a phrase from Armstrong. If a fact just is a mereological complex, then it is an “ontological free lunch,” to employ yet another signature phrase of the late Australian. Of course, not just any old mereological sum is a fact; only those with the right constituents.

And the same goes for properties: all properties, just in virtue of existing, qualify their bearers. There is no need for a tertium quid such as an instantiation relation to tie a property to its bearer. Nor is there any need for monadic facts as entities in addition to their constituents to do this unifying work. There is no difference between the sum a + F-ness and the fact of a's being F. For this to work, all properties have to be “bearer-specific.” “A property is bearer-specific if and only if it is in its nature to be had by specific bearers.” (90) Suppose it is true that Hargle is happy, and that being happy is “bearer-specific.” We can display the property as follows: __(H) being happy. '__' indicates that the property is unsaturated or incomplete or gappy in something like Frege's sense: if it is had by an individual it is had directly without the need of a connector such as an instantiation relation or Strawsonian nonrelational tie or a Bergmannian nexus. '(H)' indicates that the property is bearer-specific or rather bearer-individuated: if the property is had, it is had by Hargle and nothing else. That the property is had follows from its existence: necessarily, if the property exists, then it is had, had by Hargle and nothing else, and had directly without the service of a tertium quid. What this all implies is that the mereological sum Hargle + __(H) being happy suffices as truthmaker of 'Hargle is happy.' There is no need for a fact over and above this sum. Indeed, as Betti points out, the property alone suffices as truthmaker since it cannot exist unless Hargle exists. (101)

Questions and Objections

1. Why is Betti's proposal superior to Armstrong's?

Betti presents us with an alternative way of thinking about truthmaking facts, namely, as mereological sums whose parts include relata-specific relations and bearer-specific properties. Betti's main point is that “mereological complexes are viable as truthmakers; facts are not needed for the role.” (101) When she says that facts are not needed, she means Armstrongian, middle-period facts. She is not denying that there are truthmakers. Nor is she is denying the existence of facts as long as they are assayed as mereological complexes. If a fact is a complex entity that functions as a truthmaker, then her mereological complexes containing relata-specific relations and bearer-dependent properties are facts, though not in Armstrong 's robust sense. She is denying, or rather refusing to countenance on grounds of theoretical economy, facts as unmereological complexes. Her claim is that there is no explanatory need for facts as the middle-period Armstrong conceives of them, namely, as “additions to being.” Betti may bristle at my use of 'facts' in describing her position but surely there is an innocuous and nearly datanic, as opposed to theoretical, use of 'fact' according to which an individual's having a property, or two or more things standing in a relation, is a fact. Indeed, she needs this use of 'fact' just to state her theory, according to which the fact aRb is identical to the sum a + R + b, when R is relata-specific. On her view facts are a proper subset of mereological sums. That is not a denial of facts, but an acceptance of them. Unfortunately, Betti sometimes expresses herself in a misleading way. She tells us, for example, that “the thought that the world is a world without facts – one in which there is no difference between facts and sums – is shown to be perfectly sensible.” (88) This formulation equivocates on 'fact.' What she wants to say is that the world is without Armstrongian facts, not that the world is without truthmaking facts. It is the latter that are no different from sums, namely those sums whose constituents include relata-specific relations and object-dependent properties.

Betti thinks her theory is preferable to Armstrong's. I question whether she is justified in this preference. We face a tough choice. Armstrong's theory violates the extensionality of parthood and countenances unmereological complexes. This is a strike against it. Betti's theory avoids unmereological complexes, thereby upholding the extensionality of parthood, but accepts relata-specific relations and bearer-dependent properties. How plausible is it that all relations are relata- specific and all properties bearer-dependent? Are these notions even coherent? Let's consider the coherence question.

2. Against Relata-Specific Relations and Bearer-Dependent Properties

Suppose Argle is two feet from Bargle. There is nothing in the nature of either relatum to necessitate their standing in this external relation. Each can exist apart from the relation. And as I see it, there cannot be anything in the nature of the relation itself to necessitate that it be precisely these two critters that the relation relates. So on my view a relational situation such as Argle's being two feet from Bargle involves a double externality: there is nothing in the nature of the terms to dictate their standing in the external relation in question, and there is nothing in the nature of the external relation to dictate the terms. But as Betti sees it, it is the nature of this relation to relate Argle and Bargle and nothing else: the relation cannot exist/be instantiated without relating precisely these two. This implies that “as soon as” (105) the relation exists, it relates Argle and Bargle. If this conception is coherent, it has the desired consequence of undercutting Bertrand Russell's distinction between actually relating relations and those same relations as inert, and with it the distinction between a fact as a real unity of fact-appropriate constituents and the 'mere' mereological sum of those very same constituents. If this works, it puts paid to Armstrong's commitment to unmereological complexes: mereology suffices for truthmakers provided the parts of the sums include relata-specific relations or bearer-dependent properties.

It seems to me, however, that the notion of relata-specificity reduces to absurdity by way of the following argument in which R is any relata-specific dyadic external relation, and a and b are its individual relata. (See also my critique of D. W. Mertz in Vallicella 2004.) Generalization beyond the dyadic case is straightforward but unnecessary. Betti's definition of 'external relation' is standard and perfectly serviceable: “A relation is external if and only if it is not grounded in corresponding properties of its relata, that is, is an entity over and above its relata.” (89) An internal relation is then one that is grounded in corresponding properties and is not an entity in addition to its relata. Now to the argument:

P1. R is entirely dependent for its existence on both a and b. (Betti's theory of relata-specificity)

This is because (i) R cannot exist without being instantiated and thus cannot exist without actually relating some pair of individuals or other, and (ii) R cannot, as relata-specific, relate any pair of individuals other than a, b. If dyadic R were an immanent universal, then it could not exist without relating some pair or other; but it would not necessarily have to relate the precise pair, a, b. R's existence would then not depend on its relating a and b. But as it is, R is a particular (an unrepeatable), not a universal (a repeatable); it is a non-transferable relational trope. It is as particular as the particulars it relates. Its being or existence is exhausted by its particular occurrence, unlike an immanent universal the being or existence of which is not exhausted by its instantiation in a particular case. So R, as a relational trope, is entirely dependent for its existence on the exact relata it has: its being or existence is exhausted by its relating of those exact relata, the individuals a and b. Therefore,

C1. R is not distinct in reality from the particular relatedness aRb: R = aRb.

Of course, R can be thought of in abstraction from aRb. But R in reality is identical to aRb. You cannot say that they are different because aRb has constituents a, b while R does not. For R exists when and only when it is relating a and b. Apart from them it is nothing at all.

P2. The particular relatedness or relational fact aRb is identical to the mereological sum a + R + b, given that R is relata-specific. (Betti's theory) Therefore,

C2. R is identical to the sum a + R + b. (from C1 and P2 by Transitivity of Identity)

P3. No proper part of a mereological sum having two or more members is identical to the sum of which it is a proper part. (Principle of mereology) Therefore,

C3. R is not identical to the sum a + R + b. (from P3) Therefore,

C4. R is and is not identical to the sum a + R + b. (from C2, C3) Contradiction! Therefore,

C5. Either P1 or P2 is false; either way, Betti's theory fails.

Betti will presumably reject (C1). But how? She tells us that it is the nature of R to relate exactly a and b. Now if it is the nature of R to relate exactly these relata, then it is intrinsic to R that it do so. But then R is intrinsically relational, relational in and of itself. If this is neither contradictory nor magical, then it involves importing mind (intentionality) into the bowels of R. For if it is intrinsic to R that it relate exactly a and b, then R, quite apart from actually relating a and b, 'pre-selects' a and b as its relata. But this is what mind in its intentional states does. Such states are intrinsically relational: it is their nature to be of or about items that need not exist for the states to be of or about them. But surely there is no intentionality within the non-transferable relational trope R!

But what is the alternative? Will we be told that a and b are constituents of R? But then R is identical to aRb, when it cannot be given that aRb is a + R + b.

Now let's consider bearer-dependent properties. Suppose we grant, along with Armstrong (2004, 49), that some mereological complexes are truthmakers. Is it not also the case that some are not? Suppose that Gargle is lachrymose but Hargle is not. Then the following sum exists: Hargle + __(G)being lachrymose. The sum exists because its two parts exist. But the parts are not connected to form a truthmaker. This implies that on Betti's account there are two sorts of mereological sum: those that are truthmakers and those that are not. It also implies that what makes a mereological sum a truthmaker is not its being a mereological sum. What makes a sum a truthmaker is the nature of its members. Thus what makes Hargle + __(H)being happy a truthmaking sum is its second member.

But this second member has a rather intricate and puzzling structure. It is a bearer-individuated property, a property that exists only if instantiated by Hargle. Hargle can exist without being happy, but the property in question cannot exist unless Hargle exists. It is in the nature of the property to qualify precisely Hargle “as soon as it exists,” (105) i.e., as soon as the property exists. But when does it exist? When Hargle instantiates it. So it is not as if the property has its individuated nature apart from its being instantiated; rather, it receives its individuated nature by being instantiated by Hargle. It is only the existing Hargle that can make the property individuative of precisely Hargle and nothing else. So Hargle supplies the nature that makes the property Hargle-specific, or rather Hargle-individuated.

Does this not smack of absurdity? The nature of an entity is intrinsic to it; it cannot consist in a relation to an item external to it. So it cannot be instantiation by Hargle that gives the property its nature. If, on the other hand, Hargle were a constituent of the property in question, namely, __(H)being happy, then it would make sense to say that it is the nature of the property to be instantiated by Hargle. But Hargle is not a constituent of the property; otherwise the property would not be a property but the fact of Hargle's being happy.

Betti seems to face a dilemma. Either Hargle is not a constituent of the property or he is. If Hargle is not a constituent of the property, then the property has no nature that makes it dependent on precisely Hargle and nothing else. But if Hargle is a constituent of the property, then the property is a fact.

If Betti's account is incoherent, as I have just argued that it is, then it cannot be superior to Armstrong's even if Armstrong's is also incoherent. I should make it clear that I am not defending Armstrong; I admit that his view of facts is problematic. In fact, I argue that it is incoherent in Vallicella 2016. My point is that Betti's theory is not an acceptable replacement for it. Even if her theory is not incoherent, it is problematic as I will now further demonstrate.

3. Digging Deeper: Further Questions about Betti's Theory of Relations

Betti faults me (92-93) for failing to distinguish between externality and relata-unspecificity. A relation is external just in case it is not “grounded in corresponding properties of its relata . . . .” (89) “A rela tion is relata-unspecific if and only if it is not in its nature to relate specific relata.” (90) I fail to distinguish externality from relata-unspecificity in that I hold that, in Betti's words, “A relation is external if and only if it could have related another pair (or triple, quadruple, etc.) of relata.” (93, citing Vallicella 2002, 14-15, 31; 2004, 164). As I see it, no external relation has a nature that dictates that it relate only a particular pair, triple, quadruple, etc. of relata. As against this, Betti envisages the following possibility: an external relation such as being two feet from that holds, if it holds at all, between Argle and Bargle but cannot hold between any other pair of relata. The relation is external in that there is nothing in the natures of the relata that dictates that they stand in the relation in question; the relation is relata-specific in that there is something in the nature of the relation to dictate that, if it holds, it holds only between Argle and Bargle.

Now if Betti's scenario is possible, then I have blundered by conflating externality and relata-unspecificity. But while I grant that Betti's 'possibility' is combinatorially possible given her definitions, it is not metaphysically possible. I gave an argument above. So my conflation of externality and relata-unspecificity strikes me as justified.

I found Betti's theory of relata-specific relations (which draws on the work of her student Jan Willem Wieland) obscure and in need of further development. One intriguing suggestion is that “relata-specific relations can still be universals.” (91) Now there is a wholly uncontroversial sense of 'relata-specific universal' which Betti does not intend. Consider the universal taller than. This is a dyadic relation that is instantiated by ordered pairs of objects, but not just by any old pair. The pairs must be pairs of things having height. Taller than is thus specific to all and only such pairs and not to pairs of numbers or pairs of sets or pairs of propositions or pairs of angels or pairs of acts of thinking. But Betti means something different. She is apparently envisaging the possibility of a relation that is universal but that, say, relates only Guido, Francesca, Giacomo, and Maria in respect of height. Unfortunately, she gives no exemples and I am not sure what she is driving at. She brings this up because she thinks that her solution to the unity problem works whether or not one assays properties as universals or as tropes. (91) But this is all very obscure and here is a lacuna that needs filling.

Conclusion

My interim verdict with respect to compositional facts is that Betti has not provided a viable mereological alternative to the admittedly untenable facts or states of affairs of Armstrong's middle period.

1It is an interesting question whether one could be an idealist and also a truthmaker theorist. Consider a Kantian who holds that phenomenal objects and events are “empirically real but transcendentally ideal” to employ a signature Kantian phrase. It seems to me that such a philosopher could maintain a need for truthmakers for some truthbearers, namely those synthetic aposteriori, and thus contingent, judgments about empirical objects and events. It seems one could combine realism about empirical truth with transcendental idealism.

Sunday, August 21, 2011

I have also been perplexed at hylomorphism's dependence on something called [prime] 'matter', for the same reason as you give. But I think there is a way out, though perhaps not one a hylomorphist will like. You say "Something bare of determinateness is unthinkable and hence nonexistent." But I can think of three words that refer to something one might consider real yet bare of determinateness, namely mass (or energy), consciousness (considered apart from all intentional objects of consciousness), and God (of classical theism). In each case you have something that can be thought of as giving form actuality. But that leads to an inversion of hylomorphism, namely, that now it is form that is potential, and what was formally [formerly?] thought of as matter is now Pure Act. For example, a mathematical object which is not being thought of is a potential form that consciousness gives actuality as a thought. [. . .]

The reader is right to point out that there is something dubious about my claim that "Something bare of determinateness is unthinkable and hence nonexistent." Of the three counterexamples he gives, the clearest and best is "consciousness considered apart from all intentional objects of consciousness." Consciousness so considered is not nothing, and yet it is indeterminate since all determinations fall on the side of the objects. Consciousness is no-thing, a Sartrean theme which is also developed by Butchvarov.

The reader has made me see that there is a certain structural analogy between prime matter and consciousness conceived of as pure of-ness bare of all determinacy. For one thing, both, considered in themselves, are indeterminate or formless, and necessarily so. If consciousness were determinate, it would be an object of consciousness and not the consciousness without which there are no (intentional) objects. And if prime matter were determinate, it would be formed matter and thus not prime matter. Second, neither can exist apart from its other. There is no consciousness without objects, and there is no prime matter that exists on its own in the manner of a substance. So, while consciousness is other than every object, it cannot exist except as the consciousness of objects (objective genitive). And while prime matter is other than every form, and in itself formless, it requires formation to be something definite and substantial.

A third point of analogy is that both consciousness and prime matter give rise to a structurally similar puzzle. Consider a mind-independent hylomorph A whose matter (H) is prime matter and whose form (F) is composed of lowest forms. Which is ontologically prior, A, or its ontological parts H and F? If the parts are prior in the manner of pre-existing ontological building blocks -- think (by analogy) of the way the stones in a stone wall are prior to the wall -- then H could not be a 'principle' in the scholastic sense but would have to something capable of independent existence. And that is unacceptable: surely prime matter cannot exist on its own. If, on the other hand, A is prior to its parts, then the parts would exist only for us, or in our consideration, as aspects which we bring to A. But that won't do either because A ex hypothesi exists extramentally and so cannot in its ontological constitution require any contribution from us.

The consciousness puzzle is similar. Is consciousness (conceived as pure diaphanous of-ness of objects in the manner of Sartre, Butchvarov, and perhaps Moore) something really existent in itself or is it rather an abstract concept that we excogitate? In other words, when we think of consciousness transcendentally as the sheer revelation of objects, are we thinking of a really existent condition of their revelation, or is consciousness so conceived merely a concept that we bring to the data? If consciousness really exists, then we substantialize it (reify it, hypostatize it) in a manner analogous to the way we substantialize prime matter when we think of its as something capable of independent existence. And that is puzzling. How can something exist that is not an object of actual or possible awareness? If, on the other hand, consciousness is not something that exists on its own but is a concept that we excogitate, then how do we account for the real fact that things are apparent to us, that things are intentional objects for us? Besides, if consciousness were a mere concept, then consciousness as a reality would be presupposed: concepts are logically subsequent to consciousness.

So the two puzzles are structurally similar.

Let us see if we can abstract the common pattern. You have a term X and a distinct term Y. The terms are introduced to make sense of a phenomenon Z. Z is the analysandum whose analysis into X and Y is supposed to generate understanding. X cannot exist without Y, hence it cannot exist on its own. The same goes for Y. The terms cannot exist without each other on pain of (i) hypostatization of each, and (ii) consequent sundering of the unity of Z. (The diremption of Z into X and Y gives rise to the ancient problem of the unity of a complex which no one has ever solved.) That the terms cannot exist without each other suggests that the unitary phenomenon Z is split into X and Y only by our thoughts such that the factoring into X and Y is our contribution. On the other hand, however, the terms or factors must be capable of some sort of existence independent of our conceptual activities if the explanation that invokes them is an explanation of a real mind-independent phenomenon.

Here is a sharper form of the common aporia. Both prime matter and pure consciousness are real. But they are also both unreal. Nothing, however, can be both real and unreal on pain of violating Non-Contradiction. How remove the contradiction without giving rise to a problem that is just as bad?

I don't say that the aporiai are insoluble, but I suspect that any solution proffered with give rise to problems of its own . . . .

Friday, August 05, 2011

Your most recent post (for which many thanks) inspired the below-expressed argument, and I was curious as to your opinion of it . . . . I think it has something behind it, but right now I feel uncertain about my examples in (2).

0. There is something curious about the relation between a proposition or declarative sentence and the terms or words that compose it: the list L ("Christ," "Judas," "betrays") clearly differs, at the very least in not having a truth value, from the sentence "Judas betrays Christ," yet nothing immediately presents itself as the ground G of this difference. One plausible candidate for G is some kind of union or togetherness amongst the members of L present in "Judas betrays Christ" and not in L itself, but this proposal is open to a serious challenge.

1. Suppose we accept Barry Miller's thesis, from "Logically Simple Propositions," that some declarative sentences have only one semantic element. His favorite such sentence is the Romanian "Fulgura," whose only constituent word translates (if I remember aright) the English "brightens," and which is interesting in requiring no actual or implied subject to form a complete sentence (like "It's raining" in English, but without the dummy subject).

2. Now, the lone word in "Fulgura" seemingly can occur outside any proposition. If, for example, someone were to ask me to recite my favorite Romanian word, or to translate "brightens" into Romanian, it would be strange to take me as telling them something false, or to have them respond "No, it isn't," upon my replying with "fulgura." There would, however, be nothing strange about the sentence "Fulgura" being false and someone telling me as much. [. . .]

3. Even in such simple sentences, therefore, there is a distinction between the sentence and the words contained therein, for one can be had without the other. But the ground of this distinction cannot be any union or togetherness among the words that enter into the sentence for the simple reason that no union or togetherness amongst items can be had without distinct items to unify or bind together. It can, therefore, be at least plausibly argued that the general ground of the difference between a sentence and its constituent words is no kind of union or togetherness.

I take Mr Mollica's basic argument to be this:

a. If there are logically simple sentences/propositions, then the problem of the unity of the sentence/proposition is not one that arises for every sentence/proposition.b. There are logically simple sentences/propositions.Therefore

c. The problem of the unity of the proposition is not one that arises for every sentence/proposition.

My response is to reject (b) while granting (a). I discussed the question of logically simple sentences/propositions with Barry Miller back in the '90s in the pages of Faith and Philosophy. My "Divine Simplicity: A New Defense (Faith and Philosophy, vol. 9, no. 4, October 1992, pp. 508-525) has an appendix entitled "Divine Simplicity and Logically Simple Propositions." Miller responded and I counter-responded in the July 1994 issue, pp. 474-481. It is with pleasure that I take another look at this issue. I will borrow freely from what I have published. (Whether this counts as plagairism, depends, I suppose, on one's views on diachronic personal identity.)

A. A logically simple proposition (LSP) is one that lacks not only propositional components, but also sub-propositional components.Thus atomic propositions are not logically simple in Miller's sense, since they contain sub-propositional parts. A proposition of the form a is F, though atomic, exhibits subject-predicate complexity.

B. Miller's examples of LSPs are inconclusive. Consider the German Es regnet ('It is raining'). As Miller correctly notes, the es is grammatical filler, and so the sentence can be pared down to Regnet, which is no doubt grammatically simple. He then argues:

Now there is no question of Regnet being a predicate; for as a proposition it has a complete sense, whereas as a predicate it could have only incomplete sense. Hence, Regnet and propositions like it seem logically simple. (Barry Miller, "Logically Simple Propositions," Analysis, vol. 34, no. 4, March 1974, p. 125.)

I find it hard to avoid the conclusion that Miller is confusing propositions with the sentences used to express them. Regnet and fulgura are grammatically simple. But it scarcely follows that the propositions they express are logically simple. What makes them one-word sentences is the fact that they express propositions; otherwise, they would be mere words. So we need a sentence-proposition distinction. But once that distinction is in place then it becomes clear that grammatical simplicity of sentence does not entail logical simplicity of the corresponding proposition.

C. It is also unclear how any intellect like ours could grasp a proposition devoid of logical parts, let alone believe or know such a proposition. To believe that it is snowing, for example, is to believe something logically complex, albeit unified, something formulatable by some such sentence as 'Snow is falling.' So even if there were logically simple propositions, they could not be accusatives of minds like ours. And if propositions are defined as the possible accusatives of propositional attitudes such as belief and knowledge, then the point is stronger still: there cannot be any logically sinple propositions.

D. So it seems to me that 'the problem of the list' or the problem of the unity of the sentence/proposition is one that pertains to every sentence/proposition. It is a problem as ancient as it is tough, and, I suspect, absolutely intractable. For a glimpse into the state of the art, I shamelessly recommend my June 2010 Dialectica article, "Gaskin on the Unity of the Proposition."

Sunday, March 13, 2011

Note to Steven Nemes: Tell me if you find this totally clear, and if not, point out what is unclear. Tell me whether you accept my overall argument.

The day before yesterday in conversation Steven Nemes presented a challenge I am not sure I can meet. I have maintained (in my book, in published articles, and in these pages) that the difference between a fact and its constituents cannot be a brute difference and must therefore have a ground or explanation. But what exactly is my reasoning?

Consider a simple atomic fact of the form, a's being F. This fact has two primary constituents, the individual a, and the monadic property F-ness, which a possesses contingently. But surely there is more to the fact than these two primary constituents, and for at least two reasons. I'll mention just one, which I consider decisive: the constituents can exist without the fact existing. The individual and the property could each exist without the former exemplifying the second. This is so even if we assume that there are no propertyless individuals and no unexemplified properties. Consider a world W which includes the facts Ga and Fb. In W, a is propertied and F-ness is exemplified; hence there is no bar to saying that both exist in W. But Fa does not exist in W. So a fact is more than its primary constituents because they can exist without it existing.

A fact is not its constituents, but those constituents unified in a particular way. Now if you try to secure fact-unity by introducing one or more secondary constituents such an exemplification relation, then you will ignite Bradley's regress. For if the constituents include a, F-ness, and EX, then you still have the problem of their unity since the three can exist without constituting a fact.

So I take it as established that a fact is more than its constituents and therefore different from its constituents. A fact is different from any one of its constituents, and also from all of them taken collectively, as a mereological sum, say. The question is: What is the ontological ground of the difference? What is it that makes them different? That they are different is plain. I want to know what makes them different. It won't do to say that one is a fact while the other is not since that simply underscores that they are different. I'm on the hunt for a difference-maker.

To feel the force of the question consider what makes two different sets different. If S1 and S2 are different sets, then it is reasonable to ask what makes them different, and one would presumably not accept the answer that they are just different, that the difference is a brute difference. Let S1 be my singleton and S2 the set consisting of me and Nemes. It would not do to say that they are just different. We need a difference-maker. In this case it is easy to specify: Nemes. He is what makes S1 different from S2. Both sets contain me, but only one contains him. Generalizing, we can say that for sets at least,

DM. No difference without a difference-maker.

So I could argue that the difference between a fact and (the sum of) its constituents cannot be a brute difference because (i) there is no difference without a difference-maker and (ii) facts, sets, and sums, being complexes, are relevantly similar. (I needn't hold that the numerical difference of two simples needs a difference-maker.) But why accept (DM) in full generality as applying to all types of wholes and parts? Perhaps the principle, while applying to sets, does not apply to facts and their constituents. How do I answer the person who argues that the difference is brute, a factum brutum, and that therefore (DM), taken in full generality, is false? As we say in the trade, one man's modus ponens is another's modus tollens.

Can I show that there is a logical contradiction in maintaining that facts and their constituents just differ? That was my strategy in the book on existence. The strategy is to argue that without an external ground of unity -- an external unifer -- one lands in a contradiction, or rather cannot avoid a contradiction. That the unifier, if there is one, must be external as opposed to internal is established by showing that otherwise a vicious infinite regress ensues of the Bradley-type. I cover this ground in my book and in articles in mind-numbing detail; I cannot go over it again here. But I will refer the reader to my 2010 Dialectica article which discusses a fascinating proposal according to which unity is constituted by an internal infinite, but nonvicious, regress. But for now I assume that the unifier, if there is one, must be external. If there is one, then the difference between a fact and its constituents cannot be brute. But why must there be a unifier?

Consider this aporetic triad:

1. Facts exist.2. A fact is its constituents taken collectively.3. A fact is not its constituents taken collectively.

What I want to argue is that facts exist, but that they are contradictory structures in the absence of an external unifier that removes the contradiction. Since Nemes agrees with me about (1), I assume it for present purposes. (The justification is via the truth-maker argument).

Note that (2) and (3) are logical contradictories, and yet each exerts a strong claim on our acceptance. I have already argued for (3). But (2) is also exceedingly plausible. For if you analyze a fact, what will you uncover? Its constituents and nothing besides. The unity of the constituents whereby it is a fact as opposed to a nonfact like a mereological sum eludes analysis. The unity cannot be isolated or located within the fact. For to locate it within the fact you would have to find it as one of the constituents. And that you cannot do.

Note also that unity is not perceivable or in any way empirically detectable. Consider a simple Bergmann-style or 'Iowa' example, a red round spot. The redness and the roundness are perceivable, and the spot is perceivable. But the spot's being red and round is not perceivable. The existence of a fact is the unity of its constituents. So what I am claiming is equivalent to claiming that existence is not perceivable, which seems right: existence is not an empirical feature like redness and roundness.

So when we consider a fact by itself, there seems to be nothing more to it than its constituents.

Each limb of the triad has a strong claim on our acceptance, but they cannot all be true as formulated. The contradiction can be removed if we ascend to a higher point of view and posit an external unifier. What does that mean?

Well, suppose there is a unifier U external to the fact and thus not identifiable with one or more of its primary or secondary constituents. Suppose U brings together the constituents in the fact-making way. U would then be the sought-for ground of the fact's unity. The difference between a fact and its constituents could then be explained by saying that the difference is due to U's 'activity': U operates on the constituents to produce the fact. Our original triad can then be replaced by the following all of whose limbs can be true:

1. Facts exist2*. A fact, considered analytically, is its constituents taken collectively.3. A fact is not its constituents taken collectively.

This triad is consistent. The limbs can all be true. And I think we have excellent reason to say that each IS true. The truthmaker argument vouches for (1). (2*) looks to be true by definition. The argumentation I gave for (3) above strikes me as well-night irresistible.

But if you accept the limbs of the modified triad, then you must accept that there is something external to facts which functions as their unifier. Difficult questions about what U is and about whether U is unique and the same for all facts remain; but that U exists is 'fallout' from the modified triad. For if each limb is true, then a fact's being more than its constituents can be accounted for only by appeal to an external unifier.

But how exactly does this show that the difference between a fact and its constituents is not a brute difference? The move from the original to the modified triad is motivated by the laudable desire to avoid contradiction. So my argument boils down to this: If the difference is brute, then we get a logical contradiction. So the difference is not brute.

But it all depends on whether or not there are facts. If facts can be reasonably denied, then my reasoning to a unifer can be reasonably rejected. But that's a whole other can of worms: the truthmaker argument.

Analytically considered, a fact is just its constituents. But holistically considered it is not. Unity eludes analysis, and yet without unities there would be nothing to analyze! Analytic understanding operates under the aegis of two distinctions: whole/part, and complex/simple. Analysis generates insight by reducing wholes to their parts, and complex parts to simpler and simpler parts, and possibly right down to ultimate simples (assuming that complexity does not extend 'all the way down.') But analysis is a onesided epistemic procedure. For again, without unities there would be nothing to analyze. To understand the being-unified of a unity therefore requires that we ascend to a point of view external to the unity under analysis.

Thursday, June 24, 2010

The current issue of Dialectica (vol. 64, no. 2, June 2010) includes a symposium on Richard Gaskin, The Unity of the Proposition (Oxford 2008). Gaskin's precis of his work is followed by critical evaluations by William F. Vallicella ("Gaskin on the Unity of the Proposition"), Manuel Garcia-Carpintero ("Gaskin's Ideal Unity"), and Benjamin Schnieder ("Propositions United: Gaskin on Bradley's Regress and the Unity of the Proposition"). The symposium concludes with Gaskin's replies ("The Unity of the Proposition: Replies to Vallicella, Schnieder, and Garcia-Carpintero").

Monday, December 07, 2009

We have been discussing the view of Thomas Aquinas according to which (i) the soul is the form of the body, and (ii) the souls of some animals, namely rational animals, are subsistent, i.e. capable of an existence independent of matter. I have registered some of my misgivings. Here is another. If our souls are subsistent forms, then why are not the souls of non-human animals also subsistent? If that in us which thinks is a life-principle and the substantial form of our bodies, and subsistent to boot, by what principled means do we not ascribe subsistent souls to all living things or at least to many non-human living things?

Aquinas discusses this question in Summa Theologica, Q. 75, art. 3. He attempts to navigate a middle course between a form of materialism according to which sense and intellect are both "referred to a corporeal principle," and the view he takes to be Platonic according to which both sense and intellect are "referred to an incorporeal principle." On the Platonic view, both sensing and understanding "belong to the soul as such." Thomas points us to Theaetetus 184 c. This is a fascinating passage which, according to Paul Natorp, anticipates Kant's transcendental unity of apperception.

In this passage, Socrates puts the following question to Theaetetus: ". . . which is more correct — to say that we see or hear with the eyes and with the ears, or through the eyes and through the ears?" Theatetus obligingly responds with through rather than with. Socrates approves of this response:

Yes, my boy, for no one can suppose that in each of us, as in a sort of Trojan horse, there are perched a number of unconnected senses which do not all meet in some one nature, the mind, or whatever we please to call it, of which they are the instruments, and with which through them we perceive the objects of sense. (Emphasis added, tr. Benjamin Jowett)

The issue here is the unity of consciousness in the synthesis of a manifold of sensory data. Long before Kant, and long before Leibniz, the Plato was well aware of the problem of the unity of consciousness. (It is not for nothing that Whitehead described Western philosophy as a series of footnotes to Plato.)

Sitting before a fire, I see the flames, feel the heat, smell the smoke, and hear the crackling of the logs. The sensory data are unified in one consciousness of a selfsame object. This unification does not take place in the eyes or in the ears or in the nostrils or in any other sense organ, and to say that it takes place in the brain is not a good answer. For the brain is a partite physical thing extended in space. If the unity of consciousness is identified with a portion of the brain, then the unity is destroyed. For no matter how small the portion of the brain, it has proper parts external to each other. Every portion of the brain, no matter how small, is a complex entity. But consciousness in the synthesis of a manifold is a simple unity. Hence the unity of consciousness cannot be understood along materialist lines.

This argument from the unity of consciousness, which of course needs to be more rigorously developed, is present in nuce in Plato in the passage cited. According to Aquinas, if this argument is sound, ". . . it follows that even the souls of brute animals are subsistent." This seems to be a correct inference.

Aquinas hopes to block the inference with the help of Aristotle's De Anima 429 a 24. Thomas gives an argument that I interpret as follows:

1. Although understanding alone is performed without a corporeal organ, as Aristotle maintains, sensation and the operations of the sensitive soul are accompanied by changes in the body at our sensory receptors.

Therefore

2. The sensitive soul has no per se operation of its own, and every operation of the sensitive soul belongs to the soul-body composite.

Therefore

3. The souls of brute animals, having no per se operations, are not subsistent.

Unfortunately, I cannot see that this is a good argument. The gist of the argument is that while there are specific corporeal organs for seeing, hearing, smelling, tasting, and touching, namely, eyes, ears, etc., there is no corporeal organ for understanding. From this it is concluded that there is a crucial difference between sensing and understanding: it is broadly logically possible that there be understanding without a body; but it is not broadly logically possible that there be sensing without a body. And since we alone among animals are rational, or capable of understanding, we alone have subsistent souls.

What this ignores is that sensing is not merely a mechanical or material process. When my cat simultaneously sees, smells, and tastes her food (or my food as she much prefers) there is a unity of consciousness in her no less than there is one in me when I lay into my food. There is no specific corporeal organ that does this unifying of sensory data, neither in me nor in my cat. The unity-of-consciousness argument against materialism can be 'run' both for man and cat. If it works for me it should work for her. So if the possibility of my disembodied existence follows from there being no physical organ that unifies my consciousness, then we get the same result for my cat, and the difference between man and brute in respect of subsistence of souls cannot be maintained.

To sum up. Thomas wants to say that men, but no brutes, have subsistent souls. This is because men, but no brutes, understand. But sensing is a form of consciousness, and consciousness cannot be understood in materialist terms. Sensing is not a mere collision of atoms in the void. Sensory consciousness, besides displaying unity across its several modalities, reveals qualia. And qualia are a well-known stumbling block to materialism. It is difficult to see why, if understanding supports the possibility of disembodied existence, sensing should not also support this possibility. There is after all only one soul which both senses and understands. The phrases 'sensitive soul' and 'intellective soul' are not to be taken to refer to distinct souls.

Wednesday, October 14, 2009

The following is my contribution to a symposium on Richard Gaskin's The Unity of the Proposition. The symposium, together with Gaskin's replies, is scheduled to appear in the December 2009 issue of Dialectica.

GASKIN ON THE UNITY OF THE PROPOSITION

William F. Vallicella

While studying Richard Gaskin’s The Unity of the Proposition (Oxford 2008), the word ‘magisterial’came repeatedly to mind. Gaskin’s mastery of the history, literature, and dialectical intricacies of the problem of the unity of the proposition in all its ramifications is in evidence on every page. More than a treatment of a particular problem, Gaskin’s book is a systematic treatise in the philosophy of language organized around a particular but centrally important problem. To my knowledge, it is the most thorough and penetrating discussion of the unity of the proposition ever to appear. The fact that Gaskin’s solution to the unity problem is set within a systematic philosophy of language contributes to the book’s depth and richness, but also makes the task of the critic difficult. In a few pages, the critic cannot properly convey the systematic underpinnings of Gaskin’s formulation of the problem and his solution to it. And when the critic evaluates, he is forced to acknowledge that he is evaluating a solution embedded in a far-flung system whose ideas are mutually reinforcing. His critical points may then appear as ‘dialectical potshots’ if he cannot, as he cannot in a few pages, bring a competing system of mutually reinforcing ideas onto the field. These caveats having been registered, I proceed to sketch Gaskin’s project and raise some questions about his formulation of the unity problem. After conceding that Gaskin has solved the problem as he understands it, I will suggest that the problem lies deeper than he recognizes, and that the linguistic idealism in which he embeds his solution is problematic.

Tuesday, December 30, 2008

Beginners are taught that a set having three members is a single thing, wholly constituted by its members but distinct from them. After this, the theological doctrine of the Trinity as "three in one" should be child's play. ("The Elusiveness of Sets," Review of Metaphysics, June 1971, p. 615)

1. A set in the mathematical (as opposed to commonsense) sense is a single item 'over and above' its members. If the six shoes in my closet form a mathematical set, and it is not obvious that they do, then that set is a one-over-many: it is one single item despite its having six distinct members each of which is distinct from the set, and all of which, taken collectively, are distinct from the set. A set with two or more members is not identical to one of its members, or to each of its members, or to its members taken together, and so the set is distinct from its members taken together, though not wholly distinct from them: it is after all composed of them and its very identity and existence depends on them.

In the above quotation, Black is suggesting that mathematical sets are contradictory entities: they are both one and many. A set is one in that it is a single item 'over and above' its members or elements as I have just explained. It is many in that it is "wholly constituted" by its members. (We leave out of consideration the null set and singleton sets which present problems of their own.) The sense in which sets are "wholly constituted" by their members can be explained in terms of the Axiom of Extensionality: two sets are numerically the same iff they have the same members and numerically different otherwise. Obviously, nothing can be both one and many at the same time and in the same respect. So it seems there is a genuine puzzle here. How remove it?

Saturday, December 13, 2008

Having recently returned from the Geneva conference on Bradley's regress, I have much to ruminate upon and digest. I'll start my ruminations with some comments on Richard Gaskin's work.

In an earlier post I suggested that we ought to make a tripartite distinction among vicious, benign (harmless), and virtuous (helpful) infinite regresses. To put it crudely, a vicious regress prevents an explanatory job from getting done; a benign regress does not prevent an explanatory job from getting done; and a virtuous regress makes a positive contribution to an explanatory job's getting done. I gave an example of a putative virtuous regress in the earlier post which example I will not repeat here. In this post I draw your attention to a second putative example from the work of Richard Gaskin, whom I was happy to meet at the Geneva conference on Bradley's Regress. Gaskin's proposal is that "Bradley's regress is, contrary to to the tradition, so far from being harmful that it is even the availability of the regress which guarantees our ability to say anything at all. Bradley's regress is the metaphysical ground of the unity of the proposition." ("Bradley's Regress, the Copula, and the Unity of the Proposition," The Philosophical Quarterly, vol. 45, no. 179, April 1995, p. 176) In terms of my schema above, Gaskin is claiming that Bradley's regress is positively virtuous (not merely benign) in that it plays a positive explanatory role: it explains (metaphysically grounds) the unity of the proposition.

I will now attempt to summarize and evaluate Gaskin's position on the basis of two papers of his that I have read, and on the basis of his presentation in Geneva. (I should say that he has just published a book, The Unity of the Proposition, which I have not yet secured, so the following remarks may need revision in light of his later work.)